Solve by Factoring x^4-17x^2+16=0

Problem

x4−17*x2+16=0

Solution

1. Substitute a new variable to treat the equation as a quadratic. Let u=x2

u2−17*u+16=0

2. Factor the quadratic expression by finding two numbers that multiply to 16 and add to −17 These numbers are −16 and −1

(u−16)*(u−1)=0

3. Back-substitute x2 for u to return to the original variable.

(x2−16)*(x2−1)=0

4. Factor each term further using the difference of squares formula, a2−b2=(a−b)*(a+b)

(x−4)*(x+4)*(x−1)*(x+1)=0

5. Apply the zero product property by setting each individual factor equal to zero.

x−4=0⇒x=4

x+4=0⇒x=−4

x−1=0⇒x=1

x+1=0⇒x=−1

Final Answer

x={−4,−1,1,4}

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